The epsilon alternating least squares ($\epsilon$-ALS) is developed and analyzed for canonical polyadic decomposition (approximation) of a higher-order tensor where one or more of the factor matrices are assumed to be columnwisely orthonormal. It is shown that the algorithm globally converges to a KKT point for all tensors without any assumption. For the original ALS, by further studying the properties of the polar decomposition, we also establish its global convergence under a reality assumption not stronger than those in the literature. These results completely address a question concerning the global convergence raised in [L. Wang, M. T. Chu and B. Yu, \emph{SIAM J. Matrix Anal. Appl.}, 36 (2015), pp. 1--19]. In addition, an initialization procedure is proposed, which possesses a provable lower bound when the number of columnwisely orthonormal factors is one. Armed with this initialization procedure, numerical experiments show that the $\epsilon$-ALS exhibits a promising performance in terms of efficiency and effectiveness.

中文翻译：

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正交低秩张量逼近的 Epsilon 交替最小二乘法及其全局收敛

epsilon 交替最小二乘法 ($\epsilon$-ALS) 被开发和分析用于高阶张量的规范多元分解（近似），其中一个或多个因子矩阵被假定为列正交。结果表明，该算法在没有任何假设的情况下全局收敛到所有张量的 KKT 点。对于原始的ALS，通过进一步研究极分解的性质，我们还在一个不强于文献中的现实假设下建立了它的全局收敛性。这些结果完全解决了 [L. Wang，MT Chu 和 B. Yu，\emph {SIAM J. Matrix Anal。应用}，36 (2015)，第 1--19 页]。此外，还提出了一个初始化程序，当列正交因子数为 1 时，它具有可证明的下界。有了这个初始化程序，数值实验表明 $\epsilon$-ALS 在效率和有效性方面表现出有希望的性能。